Randomness for computable measures and initial segment complexity

TL;DR

This paper investigates the growth rates of initial segment complexity of sequences random with respect to computable measures, revealing bounds, limitations, and properties of such sequences and their Turing degrees.

Contribution

It establishes characterizations of proper sequences with bounded complexity, shows limitations of uniform results, constructs sequences with extremely slow complexity growth, and explores their Turing degrees.

Findings

Proper sequences with bounded complexity are exactly those random w.r.t. some computable, continuous measure.

A uniform bound on complexity growth does not hold across all such sequences and measures.

Existence of proper sequences with initial segment complexity below all computable functions.

Abstract

We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on $2^\omega$, the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence $X$ is bounded from below by a computable function (that is, $X$ is complex) if and only if $X$ is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure $\mu$, some sequence in this family fails to be random with respect to $\mu$. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on $2^\omega$, namely diminutive measures and trivial measures.